- SUMMARY OF RESULTS AND OPEN PROBLEMS ON RICCI SOLITONS 51
10.1. Gradient Ricci solitons on surfaces. The following is a com-
pendium of known results about complete 2-dimensional gradient Ricci soli-
tons with bounded curvature (shrinkers, steadies, and expanders). In proving
them, one may use the fact that if (M^2 , g) admits a nontrivial Killing vec-
tor field X which vanishes at some point 0 E :E, then (M, g) is rotationally
symmetric.
(1) A shrinker has constant positive curvature. In particular, the un-
derlying surface is compact.
(2) A steady is either flat or the cigar.(3) A compact expander has constant negative curvature (if x (M) < 0,
then there are no nonzero conformal Killing vector fields).
(4) An expander with positive curvature is rotationally symmetric and
unique up to homothety (see [241] and [111]).
In part ( 4), one can apply the arguments of subsection 3.1 of this chapter
and the following result (see [286] and [111]).THEOREM 1.85. If (Mn, g) is a gradient Ricci soliton on a noncom pact
manifold with Rij ~ cRgij for some E > 0, where R ~ 0, then R decays
exponentially in distance to a fixed origin. In particular, if (M^2 , g) is an
expanding gradient Ricci soliton on a surface, then R decays exponentially.A complete proof, using a more direct method, is given in [241].PROBLEM 1.86. Are there any other expanders on a surface diffeomor-
phic to IR.^2 besides the positively curved rotationally symmetric expander
and the hyperbolic disk?
PROBLEM 1.87. Do all complete 2-dimensional gradient Ricci solitons
have bounded curvature? Are all complete 2-dimensional Ricci solitons gra-
dient?
Note that in dimension 3 there are complete homogeneous expanding
solitons which are not gradient; see Section 5 of this chapter.
10.2. Gradient Ricci solitons on 3-manifolds. In dimension 3 we
have the following results. This subsection is abbreviated since we pose some
more problems in the next subsection for dimensions at least 3.
(1) (Perelman) Any nonflat shrinker with bounded nonnegative sec-
tional curvature is isometric to either a quotient of the 3-sphere or
a quotient of S^2 x R In particular, any shrinker with bounded pos-
itive sectional curvature is isometric to a shrinking solution with
constant positive sectional curvature.
(2) There exists a rotationally symmetric steady with positive sectional
curvature, namely the Bryant solitori.
(3) There exists a rotationally symmetric expander with positive sec-
tional curvature.